Chapter 7
Linear Programming Models: Graphical and Computer Models - Dr. Samir Safi
TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.
1) In the term linear programming, the word programming comes from the phrase "computer
programming."
1)
2) Any linear programming problem can be solved using the graphical solution procedure.
2)
3) An LP formulation typically requires finding the maximum value of an objective while
simultaneously maximizing usage of the resource constraints.
3)
4) There are no limitations on the number of constraints or variables that can be graphed to solve an
LP problem.
4)
5) Resource restrictions are called constraints.
5)
6) The set of solution points that satisfies all of a linear programming problem's constraints
simultaneously is defined as the feasible region in graphical linear programming.
6)
7) An objective function is necessary in a maximization problem but is not required in a minimization
problem.
7)
8) The solution to a linear programming problem must always lie on a constraint.
8)
9) In a linear program, the constraints must be linear, but the objective function may be nonlinear.
9)
10) Sensitivity analysis enables us to look at the effects of changing the coefficients in the objective
function, one at a time.
10)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) Which of the following is not a property of all linear programming problems?
A) the presence of restrictions
B) optimization of some objective
C) a computer program
D) alternate courses of action to choose from
E) usage of only linear equations and inequalities
1)
2) A feasible solution to a linear programming problem
A) must be a corner point of the feasible region.
B) must satisfy all of the problem's constraints simultaneously.
C) need not satisfy all of the constraints, only the non-negativity constraints.
D) must give the maximum possible profit.
E) must give the minimum possible cost.
2)
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3) Infeasibility in a linear programming problem occurs when
A) there is an infinite solution.
B) a constraint is redundant.
C) more than one solution is optimal.
D) the feasible region is unbounded.
E) there is no solution that satisfies all the constraints given.
3)
4) In a maximization problem, when one or more of the solution variables and the profit can be made
infinitely large without violating any constraints, the linear program has
A) an infeasible solution.
B) an unbounded solution.
C) a redundant constraint.
D) alternate optimal solutions.
E) None of the above
4)
5) Which of the following is not a part of every linear programming problem formulation?
A) an objective function
B) a set of constraints
C) non-negativity constraints
D) a redundant constraint
E) maximization or minimization of a linear function
5)
6) When appropriate, the optimal solution to a maximization linear programming problem can be
found by graphing the feasible region and
A) finding the profit at every corner point of the feasible region to see which one gives the
highest value.
B) moving the isoprofit lines towards the origin in a parallel fashion until the last point in the
feasible region is encountered.
C) locating the point that is highest on the graph.
D) None of the above
E) All of the above
6)
7) The mathematical theory behind linear programming states that an optimal solution to any
problem will lie at a(n) ________ of the feasible region.
A) interior point or center
B) maximum point or minimum point
C) corner point or extreme point
D) interior point or extreme point
E) None of the above
7)
8) Which of the following is not a property of linear programs?
A) one objective function
B) at least two separate feasible regions
C) alternative courses of action
D) one or more constraints
E) objective function and constraints are linear
8)
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9) Consider the following linear programming problem:
Maximize
Subject to:
9)
12X + 10Y
4X + 3Y ≤ 480
2X + 3Y ≤ 360
all variables ≥ 0
The maximum possible value for the objective function is
A) 360.
B) 480.
C) 1520.
D) 1560.
E) None of the above
10) Consider the following linear programming problem:
Maximize
Subject to:
10)
4X + 10Y
3X + 4Y ≤ 480
4X + 2Y ≤ 360
all variables ≥ 0
The feasible corner points are (48,84), (0,120), (0,0), (90,0). What is the maximum possible value for
the objective function?
A) 1032
B) 1200
C) 360
D) 1600
E) None of the above
11) Consider the following linear programming problem:
Maximize
Subject to:
5X + 6Y
4X + 2Y ≤ 420
1X + 2Y ≤ 120
all variables ≥ 0
Which of the following points (X,Y) is not a feasible corner point?
A) (0,60)
B) (105,0)
C) (120,0)
D) (100,10)
E) None of the above
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11)
12) Consider the following linear programming problem:
Maximize
Subject to:
12)
5X + 6Y
4X + 2Y ≤ 420
1X + 2Y ≤ 120
all variables ≥ 0
Which of the following points (X,Y) is not feasible?
A) (50,40)
B) (20,50)
C) (60,30)
D) (90,10)
E) None of the above
13) Two models of a product — Regular (X) and Deluxe (Y) — are produced by a company. A linear
programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60 Y
Subject to:
8X + 10Y ≤ 800
X + Y ≤ 120
4X + 5Y ≤ 500
all variables ≥ 0
13)
(labor hours)
(total units demanded)
(raw materials)
The optimal solution is X = 100, Y = 0.
How many units of the regular model would be produced based on this solution?
A) 120
B) 0
C) 100
D) 50
E) None of the above
14) Which of the following is not acceptable as a constraint in a linear programming problem
(maximization)?
Constraint 1
Constraint 2
Constraint 3
Constraint 4
14)
X + XY + Y ≥ 12
X - 2Y ≤ 20
X + 3Y = 48
X + Y + Z ≤ 150
A) Constraint 1
B) Constraint 2
C) Constraint 3
D) Constraint 4
E) None of the above
15) Sensitivity analysis may also be called
A) postoptimality analysis.
B) optimality analysis.
C) parametric programming.
D) All of the above
E) None of the above
15)
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16) If the addition of a constraint to a linear programming problem does not change the solution, the
constraint is said to be
A) bounded.
B) infeasible.
C) redundant.
D) non-negative.
E) unbounded.
16)
17) The difference between the left-hand side and right-hand side of a less-than-or-equal-to
constraint is referred to as
A) slack.
B) surplus.
C) constraint.
D) shadow price.
E) None of the above
17)
18) In order for a linear programming problem to have a unique solution, the solution must exist
A) at the intersection of two or more constraints.
B) at the intersection of a non-negativity constraint and a resource constraint.
C) at the intersection of the non-negativity constraints.
D) at the intersection of the objective function and a constraint.
E) None of the above
18)
19) Consider the following linear programming problem:
19)
Maximize
Subject to:
12X + 10Y
4X + 3Y ≤ 480
2X + 3Y ≤ 360
all variables ≥ 0
Which of the following points (X,Y) is feasible?
A) (120,10)
B) (30,100)
C) (10,120)
D) (60,90)
E) None of the above
20) In order for a linear programming problem to have multiple solutions, the solution must exist
A) on a non-redundant constraint parallel to the objective function.
B) at the intersection of three or more constraints.
C) at the intersection of the non-negativity constraints.
D) at the intersection of the objective function and a constraint.
E) None of the above
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20)
21) Consider the following linear programming problem:
Maximize
Subject to:
21)
5X + 6Y
4X + 2Y ≤ 420
1X + 2Y ≤ 120
all variables ≥ 0
Which of the following points (X,Y) is in the feasible region?
A) (30,60)
B) (100,10)
C) (105,5)
D) (0,210)
E) None of the above
22) Which of the following is not acceptable as a constraint in a linear programming problem
(minimization)?
Constraint 1
Constraint 2
Constraint 3
Constraint 4
Constraint 5
22)
X + Y ≥ 12
X - 2Y ≤ 20
X + 3Y = 48
X + Y + Z ≤ 150
2X - 3Y + Z > 75
A) Constraint 1
B) Constraint 2
C) Constraint 3
D) Constraint 4
E) Constraint 5
23) Consider the following constraints from a linear programming problem:
23)
2X + Y ≤ 200
X + 2Y ≤ 200
X, Y ≥ 0
If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?
A) (65, 65)
B) (100, 0)
C) (0, 0)
D) (66.67, 66.67)
E) (0, 100)
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
1) A furniture company is producing two types of furniture. Product A requires 8 board feet of wood and 2 lbs of
wicker. Product B requires 6 board feet of wood and 6 lbs of wicker. There are 2000 board feet of wood available
for product and 1000 lbs of wicker. Product A ea rns a profit margin of $30 a unit and Product B earns a profit
margin of $40 a unit. Formulate the problem as a linear program.
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2) As a supervisor of a production department, you must decide the daily production totals of a certain product
that has two models, the Deluxe and the Special. The profit on the Deluxe model is $12 per unit and the
Special's profit is $10. Each model goes through two phases in the production process, and there are only 100
hours available daily at the construction stage and only 80 hours available at the finishing and inspection stage.
Each Deluxe model requires 20 minutes of construction time and 10 minutes of finishing and inspection time.
Each Special model requires 15 minutes of construction time and 15 minutes of finishing and inspection time.
The company has also decided that the Special model must comprise at least 40 percent of the production total.
(a) Formulate this as a linear programming problem.
(b) Find the solution that gives the maximum profit.
3) The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and
liver-flavored biscuits) that meets certain nutritional requirements. The liver-flavored biscuits contain 1 unit of
nutrient A and 2 units of nutrient B, while the chicken-flavored ones contain 1 unit of nutrient A and 4 units of
nutrient B. According to federal requirements, there must be at least 40 units of nutrient A and 60 units of
nutrient B in a package of the new biscuit mix. In addition, the company has decided that there can be no more
than 15 liver-flavored biscuits in a package. If it costs 1 cent to make a liver-flavored biscuit and 2 cents to
make a chicken-flavored one, what is the optimal product mix for a package of the biscuits in order to
minimize the firm's cost?
(a) Formulate this as a linear programming problem.
(b) Find the optimal solution for this problem graphically.
(c) Are any constraints redundant? If so, which one or ones?
(d) What is the total cost of a package of dog biscuits using the optimal mix?
4) Consider the following linear program:
Maximize
30X 1 + 10X2
Subject to:
3X1 + X2 ≤ 300
X1 + X2 ≤ 200
X1 ≤ 100
X2 ≥ 50
X 1 ≥ X2 ≤ 0
X1 , X2 ≥ 0
(a) Solve the problem graphically. Is there more than one optimal solution? Explain.
(b) Are there any redundant constraints?
5) Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day. One of
these is the Standard model, while the other is the Deluxe model. The profit per unit on the Standard model is
$60, while the profit per unit on the Deluxe model is $40. The Standard model requires 20 minutes of assembly
time, while the Deluxe model requires 35 minutes of assembly time. The Standard model requires 10 minutes
of inspection time, while the Deluxe model requires 15 minutes of inspection time. The company must fill an
order for 6 Deluxe models. There are 450 minutes of assembly time and 180 minutes of inspection time
available each day. How many units of each product should be manufactured to maximize profits?
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6) Susanna Nanna is the production manager for a furniture manufacturing company. The company produces
tables (X) and chairs (Y). Each table generates a profit of $80 and requires 3 hours of assembly time and 4 hours
of finishing time. Each chair generates $50 of profit and requires 3 hours of assembly time and 2 hours of
finishing time. There are 360 hours of assembly time and 240 hours of finishing time available each month.
The following linear programming problem represents this situation.
Maximize
Subject to:
80X + 50Y
3X + 3Y ≤ 360
4X + 2Y ≤ 240
X, Y ≥ 0
The optimal solution is X = 0, and Y = 120.
(a) What would the maximum possible profit be?
(b) How many hours of assembly time would be used to maximize profit?
(c) If a new constraint, 2X + 2Y ≤ 400, were added, what would happen to the maximum possible profit?
7) Consider the following constraints from a two-variable linear program.
(1) X ≥ 0
(2) Y ≥ 0
(3) X + Y ≤ 50
If the optimal corner point lies at the intersection of constraints (2) and (3), what is the optimal solution (X, Y)?
8) Consider a product mix problem, where the decision involves determining the optimal production levels for
products X and Y. A unit of X requires 4 hours of labor in department 1 and 6 hours a labor in department 2. A
unit of Y requires 3 hours of labor in department 1 and 8 hours of labor in department 2. Currently, 1000 hours
of labor time are available in department 1, and 1200 hours of labor time are available in department 2.
Furthermore, 400 additional hours of cross-trained workers are available to assign to either department (or split
between both). Each unit of X sold returns a $50 profit, while each unit of Y sold returns a $60 profit. All units
produced can be sold. Formulate this problem as a linear program. (Hint: Consider introducing other decision
variables in addition to the production amounts for X and Y.)
9) A company can decide how many additional labor hours to acquire for a given week. Subcontractor workers
will only work a maximum of 20 hours a week. The company must produce at least 200 units of product A, 300
units of product B, and 400 units of product C. In 1 hour of work, worker 1 can produce 15 units of product A,
10 units of product B, and 30 units of product C. Worker 2 can produce 5 units of product A, 20 units of product
B, and 35 units of product C. Worker 3 can produce 20 units of product A, 15 units of product B, and 25 units of
product C. Worker 1 demands a salary of $50/hr, worker 2 demands a salary of $40/hr, and worker 3 demands
a salary of $45/hr. The company must choose how many hours they should contract with each worker to meet
their production requirements and minimize labor cost.
(a) Formulate this as a linear programming problem.
(b) Find the optimal solution.
10) Define unboundedness with respect to an LP solution.
11) Define infeasibility with respect to an LP solution.
12) Define alternate optimal solutions with respect to an LP solution.
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13) How does the case of alternate optimal solutions, as a special case in linear programming, compare to the two
other special cases of infeasibility and unboundedness?
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Answer Key
Testname: CHAPTER 7
1) FALSE
2) FALSE
3) FALSE
4) FALSE
5) TRUE
6) TRUE
7) FALSE
8) TRUE
9) FALSE
10) TRUE
1) C
2) B
3) E
4) B
5) D
6) A
7) C
8) B
9) C
10) B
11) C
12) A
13) C
14) A
15) D
16) C
17) A
18) A
19) B
20) A
21) B
22) E
23) A
1)
Let X 1 = number of units of Product A produced
2)
(a) Let
X2 = number of Special models produced
30X1 + 40X2
Subject to:
8X1 + 6X2 ≤ 2000
Maximize
12X1 + 10X2
Subject to:
1/3 X1 + 1/4 X2 ≤ 100
1/6 X1 + 1/4 X2 ≤ 80
-0.4X 1 + 0.6X2 ≥ 0
X 1 , X2 ≥ 0
(b) Optimal solution: X1 = 120, X2 = 240
$3,840
Profit =
3)
(a) Let
X1 = number of liver-flavored biscuits in a package
X2 = number of chicken-flavored biscuits in a packag
Minimize
X1 + 2X2
Subject to:
X1 + X2 ≥ 40
2X1 + 4X2 ≥ 60
X1 ≤ 15
X1 , X2 ≥ 0
(b) Corner points (0,40) and (15,25)
Optimal solution is (15,25) with cost of 65.
(c) 2X1 + 4X2 ≥ 60 is redundant.
(d) minimum cost = 65 cents
4) (a) Corner points (0,50), (0,200), (50,50), (75,75),
(50,150)
Optimum solutions: (75,75) and (50,150). Both
yield a profit of $3,000.
(b) The constraint X1 ≤ 100 is redundant since 3X1 +
X 2 = number of units of Product B produced
Maximize
X1 = number of Deluxe models produced
X2 ≤ 300 also means that X1 cannot exceed 100.
5)
Let
2X1 + 6X2 ≤ 1000
X 1 , X2 ≥ 0
X = number of Standard models to produce
Y = number of Deluxe models to produce
Maximize
Subject to:
60X + 40Y
20X + 35Y ≤ 450
10X + 15Y ≤ 180
Y≥6
X, Y ≥ 0
Maximum profit is $780 by producing 9 Standard
and 6 Deluxe models.
6) (a) 6000, (b) 360, (c) It would not change.
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Answer Key
Testname: CHAPTER 7
7) Y = 0, so X + 0 = 50, or X = 50. Thus the solution is
(50, 0).
8)
Let
X = the number of units of product X sold
Y = the number of units of product Y sold
C1 = the number of cross-trained labor hours allocated to department 1
C2 = the number of cross-trained labor hours allocated to department 2
Maximize: 50X + 60Y
Subject to: 4X + 3Y ≤ 1000 + C1
6X + 8Y ≤ 1200 + C2
C1 + C2 ≤ 400
X, Y ≥ 0
9)
(a) Let X1 = Worker 1 hours
X2 = Worker 2 hours
X3 = Worker 3 hours
Minimize
50X 1 + 40X2 + 45 X3
Subject to:
15X 1 + 5X 2 + 20X3 ≥ 200
10X 1 + 20X2 + 15X3 ≥ 300
30X 1 + 35X2 + 25X3 ≥ 400
X1 , X2 , X3 ≤ 20
X1 , X2 , X3 ≥ 0
(b) X1 = 0, X2 = 9.23, X3 = 7.69
10) This occurs when a linear program has no finite
solution. The result implies that the formulation is
missing one or more crucial constraints.
11) This occurs when there is no solution that can satisfy
all constraints simultaneously.
12) More than one optimal solution point exist because
the objective function is parallel to a binding
constraint.
13) With multiple alternate solutions, any of those
answers is correct. In the other two cases, no single
answer can be generated. Alternate solutions can
occur when a problem is correctly formulated
whereas the other two cases most likely have an
incorrect formulation.
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